Which of the following is the perpendicular bisector theorem?

a) if a point is equidistant from the endpoints of the segment, then it lies on the perpendicular bisector of that segment.

b) if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

c) if a point is on the perpendicular bisector of a segment, then it is proportional from the endpoints of the segment.

d) if a point is proportional from the endpoints of the segment, then it lies on the perpendicular bisector of that segment.

Solution:- Option B) is the right answer . "If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment."

Explanation:-

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the segment's both endpoints.

Option A) tells the converse of perpendicular theorem which states that if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)